Document why mod2pi(2pi) != 0

[jiahao]

julia> help(mod2pi)
Loading help data...
Base.mod2pi(x)

   Modulus after division by 2pi, returning in the range [0,2pi). More
   accurate than mod(x,2pi).

julia> 2pi
6.283185307179586

julia> mod2pi(2pi)
6.283185307179586

julia> mod(2pi, 2pi)
0.0

[ivarne]

julia> mod2pi(nextfloat(2pi))
6.432490598706546e-16

julia> big(2pi) < big(2.)pi
true

[jiahao]

Yes, clearly that was the explanation. This is a floating-point gotcha that really should be documented.

[ivarne]

Like this method is crazy good and will give the modules of the exact float you give it, and not make a wild guess for what you might have intended to ask for?

[jiahao]

Like this method is crazy good and will give the modules of the exact float you give it,

Yes, in that it computes the modulus relative to numerically exact 2pi rather than its floating-point approximation.

not make a wild guess for what you might have intended to ask for?

The point is that one can ask for a precise floating point computation that doesn't do what new users would naïvely expect it to do. The behavior typeof(pi)<:MathConst but typeof(2pi)==Float64 and yet the docstring using 2pi to mean numerically exact 2π just adds to the confusion here. Telling that them they are forcing their program to "make a wild guess" at their intentions isn't terribly constructive.

[jiahao]

I'm wondering if @StefanKarpinski was right all those months ago in wanting a tau constant.

[ivarne]

Floating point accuracy is hard to get right, and it tends to bite fairly hard. I'll use this opportunity to bump my suggestion in #4278 to depreciate big(f::Float64) and BigFloat(f::Float64) and direct users to the string versions or convert(::BigFloat, f::Float64) if they really want their almost correct number do get many more wrong bits.